Optimal Flux Densities for Linear Mappings and the Multiscale Geometry of Structured Deformations

We establish the unexpected equality of the optimal volume density of total flux of a linear vector field (Formula presented.) and the least volume fraction that can be swept out by submacroscopic switches, separations, and interpenetrations associated with the purely submacroscopic structured deformation (i, I + M). This equality is established first by identifying a dense set (Formula presented.) of (Formula presented.) matrices M for which the optimal total flux density equals |trM|, the absolute value of the trace of M. We then use known representation formulae for relaxed energies for structured deformations to show that the desired least volume fraction associated with (i, I + M) also equals |trM|. We also refine the above result by showing the equality of the optimal volume density of the positive part of the flux of (Formula presented.) and the volume fraction swept out by submacroscopic separations alone, with common value (trM)+. Similarly, the optimal volume density of the negative part of the flux of (Formula presented.) and the volume fraction swept out by submacroscopic switches and interpenetrations are shown to have the common value (trM)−.